Jump of the Milnor Number in Linear Deformations with Fixed Order

Abstract

Let $f_0$ be a plane curve singularity. We study the jump of the Milnor number of a homogeneous singularity $f_0$ of order $n$ in the class of linear deformations $f_s=f_0+sg$ of $f_0$ with fixed order $ord{f}_s=j<n$ for $s\ne 0$, where $g$ is holomorphic and $ordg=j$.